Διαφορικός Τελεστής
Διαφορικός Τελεστής Differential Operator thumb|300px| [[Ανάδελτα Εφαπτομένη Εφαπτόμενο Διάνυσμα Εφαπτομενικός Χώρος Διαφορικός Τελεστής Παράγωγος Διαφορική Γεωμετρία ]] - Ένας Τελεστής. Ετυμολογία Η ονομασία "Διαφορικός" σχετίζεται ετυμολογικά με την λέξη "Διαφορικό". Εισαγωγή Είναι ο τελεστής που εμφανίζεται ως συνάρτηση του τελεστή της διαφόρισης (δηλ. της παραγώγου). Υπάρχουν πολλά είδη διαφορικών τελεστών * η Χρονική Παράγωγος, η Χωρική Παράγωγος * η Ολική Παράγωγος, η Μερική Παράγωγος * η Κλίση (grad) * η Απόκλιση (div) * η Στροβιλισμός (curl) * η Περιστροφή (rot) In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear operators, which are the most common type. However, non-linear differential operators, such as the Schwarzian derivative also exist. Definition Assume that there is a map A from a function space \mathcal{F}_1 to another function space \mathcal{F}_2 and a function f \in \mathcal{F}_2 so that f is the image of u \in \mathcal{F}_1 i.e.,　 f=A(u)\ . A differential operator is represented as a linear combination, finitely generated by u and its derivatives containing higher degree such as : P(x,D)=\sum_{|\alpha|\le m}a_\alpha(x) D^\alpha\ , where the set of non-negative integers, \alpha=(\alpha_1,\alpha_2,\cdots,\alpha_n) , is called a multi-index, |\alpha|=\alpha_1+\alpha_2+\cdots+\alpha_n called length, a_\alpha(x) are functions on some open domain in n''-dimensional space and D^\alpha=D^{\alpha_1} D^{\alpha_2} \cdots D^{\alpha_n}\ . The derivative above is one as functions or, sometimes, distributions or hyperfunctions and D_j=-i\frac{\partial}{\partial x_j} or sometimes, D_j=\frac{\partial}{\partial x_j} . Notations The most common differential operator is the action of taking the derivative itself. Common notations for taking the first derivative with respect to a variable ''x include: : {d \over dx}, D,\, D_x,\, and \partial_x. When taking higher, n''th order derivatives, the operator may also be written: : {d^n \over dx^n}, D^n\,, or D^n_x.\, The derivative of a function ''f of an argument x'' is sometimes given as either of the following: : f(x)'\,\! : f'(x).\,\! The ''D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form : \sum_{k=0}^n c_k D^k in his study of differential equations. One of the most frequently seen differential operators is the Laplacian operator, defined by : \Delta=\nabla^{2}=\sum_{k=1}^n {\partial^2\over \partial x_k^2}. Another differential operator is the Θ operator, or theta operator, defined by : \Theta = z {d \over dz}. This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z'': : \Theta (z^k) = k z^k,\quad k=0,1,2,\dots In ''n variables the homogeneity operator is given by : \Theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}. As in one variable, the eigenspaces of Θ are the spaces of homogeneous polynomials. In writing, following common mathematical convention, the argument of a differential operator is usually placed on the right side of the operator itself. Sometimes an alternative notation is used: The result of applying the operator to the function on the left side of the operator and on the right side of the operator, and the difference obtained when applying the differential operator to the functions on both sides, are denoted by arrows as follows: : f \overleftarrow{\partial_x} g = g \cdot \partial_x f : f \overrightarrow{\partial_x} g = f \cdot \partial_x g : f \overleftrightarrow{\partial_x} g = f \cdot \partial_x g - g \cdot \partial_x f. Such a bidirectional-arrow notation is frequently used for describing the probability current of quantum mechanics. Υποσημειώσεις Εσωτερική Αρθρογραφία * Ολοκληρωτικός Τελεστής * Ανάδελτα Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *